474 karine chemla
of Mathematical Procedures contains in relation to this operation, which
provides a hint that this was how the situation was understood even before
Th e Nine Chapters was compiled. It is important in this respect that one of
the prerequisites for this interpretation – namely, that the multiplication
inverse to a division restores the original number divided – appears to be
a concern documented in the Book of Mathematical Procedures , as shown
above. Th is completes our argument in this case.
An additional remark should be made. So far, in our argument, we have
only considered the validity of transformation iii with respect to integers.
What about establishing its validity more generally? Two points should be
added in this respect.
On the one hand, if we observe the contexts in which ‘dividing at a
stroke’, or its synonym, ‘dividing together’ ( bingchu ), are used in the com-
mentaries, it turns out that the two divisions that are joined are usually both
divisions by integers.
On the other hand, if this is the case, in some situations this relates to
the fact that the property of entities to be lü was put into play. 67 S i m i l a r l y ,
if the capacity of the quantities involved to be transformed into integers is
employed, transformation iii is to be used in contexts in which they were
already turned into integers. Th at such may have been the idea is plausible:
more generally, Th e Nine Chapters exhibits a way of carrying out compu-
tations that grants a predominant part to integers, and the introduction
of the concept of lü can be interpreted as one technique among several
devised to fulfi l this aim. Several hints can be given in favour of these
hypotheses.
First, the commentators regularly interpret the choice of describing a
procedure in a given way in Th e Nine Chapters as derived from the motiva-
tion of the authors to avoid generating fractions in the midst of computa-
tions. Th is is how, for instance, Liu Hui accounts for why, in the rule of
three, the multiplication is prescribed before the division, and not aft er. 68
Th e commentators thus attribute to Th e Nine Chapters the intention of
computing with integers wherever possible.
Second, the way in which division between quantities containing frac-
tions is dealt with in the general case amounts precisely to getting rid of
fractions. Liu Hui reads this way of proceeding as made possible by the
status of the dividend and divisor as lü s. Th ird, in the procedure of Th e
Nine Chapters in the context of which the concept of lü is introduced, that
67 See, for instance, how Liu Hui interprets the algorithm provided aft er problem 6.10 (CG2004:
514–15).
68 Th e validity of this operation is discussed in the next subsection.